General duality and magnet-free passive phononic Chern insulators

Integrated phononics plays an important role in both fundamental physics and technology. Despite great efforts, it remains a challenge to break time-reversal symmetry to achieve topological phases and non-reciprocal devices. Piezomagnetic materials offer an intriguing opportunity as they break time-reversal symmetry intrinsically, without the need for an external magnetic field or an active driving field. Moreover, they are antiferromagnetic, and possibly compatible with superconducting components. Here, we develop a theoretical framework that combines linear elasticity with Maxwell’s equations via piezoelectricity and/or piezomagnetism beyond the commonly adopted quasi-static approximation. Our theory predicts and numerically demonstrates phononic Chern insulators based on piezomagnetism. We further show that the topological phase and chiral edge states in this system can be controlled by the charge doping. Our results exploit a general duality relation between piezoelectric and piezomagnetic systems, which can potentially be generalized to other composite metamaterial systems.


Section 1: Derivation of the master equation
On the other hand, the movement of doped charges creates a current which appears in Maxwell's equations as ∇ × = + , here is the electric displacement field.
These effects modify the overall ℋ to Next, we will show that by allowing complex amplitude for vector , ℬ = ℬ † for a lossless system. We only show the PZM effect while the PZE effect can be derived similarly.
Therefore, the Eq. 3 can be written as:

Section 2: A simplified model of PZM effect
PZM materials intrinsically break . [2][3][4] This can be understood from that the PZM tensor is -odd, since it links the -even strain tensor (rank two) and the -odd magnetic field vector. To

Section 3: Complex valued in the presence of general duality
At = 0, ℋ is block diagonal, this makes it possible to put a phase difference between ( , ) and ( , ). Therefore, if is pure imaginary, it can be mapped back to a real case. When is complex in generally (note this is a matrix) and cannot be reduced to a real , it usually means there are some dynamics coupled. For example, when net magnetization is induced, the precession gives both real and imaginary part in the susceptibility, making effectively complex 5 .
Generally, it can be numerically verified that if a precession is presented, the topological bandgap can be opened. However, this case is physically different from the case with real . When is real, the net magnetic moment is 0 and the -broken order parameter is the "staggered magnetization" 6 . The staggered magnetization determines (also see the simplified model in Section 2). When is complex, the order parameter is the non-zero magnetic moment. The Operator maps symmetries from one system to its dual system.
Using the unitary operator defined in the main text, we have −1 = . This makes it possible to "copy" a symmetry from to at = 0.

Section 5: The block diagonalization of the asymmetric and symmetric modes
The

Section 6: The effect of the general duality in the PZE system
As a direct result from the general duality, PZE system, which is a dual system of the PZM system as demonstrated in the main text, is also affected by the general duality. More specifically, in a PZE system, where is broken but is preserved, the general duality creates ̃=  The 2D simulation is done with the same design as Fig. 1a in the main text. Intuitively, the Dirac point can be broken by PZE terms since symmetry is also broken in this case. However, due to the presence of ̃ and thus ̃, the Dirac point degeneracy lifting cannot be done by PZE terms only (Fig. S2a) and requires the presence of (Fig. S2b). The control of the symmetrybreaking orders by is revealed in a simulation with similar structure to Fig. 2a.

Section 7: Calculation of overall charge neutral phononic Chern insulators
In Fig. 4 of the main text, we deliberately set to have both signs within the unit cell, where a negative charge "puddle" has the same charge density as the nearby positive one, and the charge distribution pattern keeps 6ℎ symmetry. Therefore, the sum of the doping charges over a unit cell is strictly zero. In this setup, the absolute value of /q 0 is set to 1.0 within the puddles, much larger than what is used in the uniform charge case. The 0 is doubled to 0.8. The super cell is set to have 20 Chern insulator unit cells and 10 trivial insulator unit cells at y direction. The material parameters of trivial insulator unit cells are the same as Fig. 1b in the main text, which puts the topological band gap at ~0.75 GHz into a trivial gap.

Section 8: Bandgap size estimation
Apart from the PZM material described in the main text. A closely-related family of materials magnetostrictive materials, operating in a modest external magnetic field of ~10 Oe, can have an effective 0 / 0 up to as large as 12 20. Assuming the gap size varies linearly with / 0 and 0 / 0 , the gap can easily become larger than 10 MHz ( 0 / 0 = 1, / 0 = 0.3 or 0 / 0 = 10, / 0 = 0.03), which is 1.5% of the working frequency, large enough compared with the overall bandwidth (in Fig. 1d, for example, the frequency range covered by band I, II and III areall three bands is <100 ~80 MHz). In this kind of material, loss is expected. By using Gilbert damping constant and other parameters in Ref. 12, and making both the and terms complex according to the ferromagnetic resonance equation, we estimate the linewidth broadening at 0.68 GHz by imaginary part of the eigenvalue solution, which is ~0.2% of the central frequency. By selecting different working frequency and doing other optimizations, we expect this loss to be further reduced.
Even larger PZM effect may be achieved using metamaterials with structures like Fig. S1.